\documentclass{PoS}

\usepackage{epsfig}

\title{Three-loop heavy quark potential\thanks{This work is supported by DFG through SFB/TR~9
          ``Computational Particle Physics'' and RFBR, grant 08-02-01451.
        Preprint Nos.: SFB/CPP-10-82, TTP10-36}}

\ShortTitle{Three-loop heavy quark potential}

\author{Alexander V. Smirnov\\
Scientific Research Computing Center of Moscow State University,
Russia\\
E-mail: \email{asmirnov@rdm.ru}}

\author{Vladimir A. Smirnov\\
Nuclear Physics Institute of Moscow State University, Russia\\
E-mail: \email{smirnov@theory.sinp.msu.ru}}

\author{\speaker{Matthias Steinhauser}%
        	   \\
        KIT\\
        E-mail: \email{matthias.steinhauser@kit.edu}}



\abstract{We discuss the calculation of the three-loop corrections to
  the static potential of two heavy quarks.}

\FullConference{35th International Conference of High Energy Physics\\
                 July 22-28, 2010\\
                 Paris, France}

\begin{document}

\section{Introduction}

The potential between two heavy quarks has been among the 
first application after the formulation of QCD. At leading order it is
given by the exchange of a Coulomb gluon and can --- after obvious
modifications --- be obtained from the potential of the hydrogen atom.
One- and two-loop corrections have been considered in
Refs.~\cite{Fischler:1977yf,Billoire:1979ih,Peter:1996ig,Peter:1997me,Schroder:1998vy,Kniehl:2001ju}
and have introduced numerically sizable 
effects in quarkonium physics (see,
e.g., the review Ref.~\cite{Brambilla:2004wf}).
Since 1998 there has been a raising interest in
the three-loop corrections. The fermionic corrections have been
completed in 2008~\cite{Smirnov:2008pn} and in 2009 two
independent groups~\cite{Smirnov:2009fh,Anzai:2009tm} have completed
the purely abelian three-loop part.
In Refs.~\cite{Smirnov:2008pn,Smirnov:2009fh} the calculation has been
performed in a covariant gauge and the independence of the final
expression on the gauge parameter has been a crucial check for the
correctness of the result.

\section{Outline of the calculation}

The calculation of the static potential requires the evaluation of the
four-point amplitude of a heavy quark and anti-quark. Some sample Feynman
diagrams are shown in Fig.~\ref{fig::diags}. It is sufficient
to consider as a starting point the so-called non-relativistic QCD
(NRQCD), i.e. QCD with hard degrees of freedom integrated out. In this
limit the heavy quark propagators represent static colour sources with
propagators $1/p_0$ whereas the gluons and light quarks are still
relativistic. The only dimensionful scale in the problem is the
momentum transfer between the heavy quark and anti-quark and thus
momentum integrals can be represented by two-point functions. In
Fig.~\ref{fig::scalar} the different cases of the 
scalar two-point integrals up to three loops are shown.

\begin{figure}[b]
  \centering
  \leavevmode
  \epsfxsize=.7\textwidth
  \epsffile[160 330 560 490]{figs/diag1_1.ps}
%%%  \includegraphics[width=\textwidth]{figs/diag1_1.ps}
%%%    \epsfig{file=figs/diag1.ps,width=\textwidth}
  \vspace*{-.5em}
  \caption{\label{fig::diags}Sample diagrams contributing to the
    static potential at tree-level, one-, two- and three-loop order.
    Solid and curly lines represent quarks and gluons, respectively.
    In the case of closed loops the quarks are massless; the external 
    quarks are heavy and treated in the static limit.
    }
\end{figure}

\begin{figure}[t]
  \centering
  \leavevmode
  \epsfxsize=.8\textwidth
  \epsffile[80 400 550 440]{figs/diag22.ps}
  \vspace*{-.5em}
  \caption{\label{fig::scalar}Scalar one-, two- and three-loop diagrams.
    The solid line stands for massless relativistic propagators and the 
    zigzag line represents static propagators.
    }
\end{figure}

In case the static lines are absent the problem of computing the
corresponding integrals up to three loops has been solved many years
ago~\cite{Chetyrkin:1981qh} and a public code exists, {\tt MINCER}~\cite{Larin:1991fz}
which can easily be included in all computational frameworks. The
presence of the static lines, however, makes the practical evaluation
quite difficult and an explicit solution of the recurrence problem (as
implemented in Ref.~\cite{Larin:1991fz}) is not available. Furthermore, the
master integrals are significantly more complicated due to the
occurrence of the static lines.

In Refs.~\cite{Smirnov:2008pn,Smirnov:2009fh} the reduction of all
occurring integrals to a small set of master integrals has been
achieved with the help of the program {\tt FIRE}~\cite{FIRE} which can
be linked to a database and thus handle non-trivial problems in a
quite efficient way. In our case up to 16 indices have to be
considered: $8+6=14$ indices from relativistiv and non-relativistic
propagators, respectively, and in addition one index from an irreducible
numerator (see Fig.~\ref{fig::scalar}). The problem can be simplified
by considering a partial fractioning in those cases where three static
lines meet in one vertex and by linear relations between four static
propagators which leads to at most three static
propagators at three loops thus reducing the total number of indices
to twelve. We have performed the calculation in both ways. In the first
option with up to 15 indices only little manual work is involved,
however, significantly more computer resources are needed than in the
twelve-index approach. In the latter case one has to provide several
relations implemeting the partial fractioning (mentioned above) and
symmetry relations to end up with a small set of different case to be
considered for the reduction. The fact that the final results in both
approaches agree constitutes a strong check on our result.

After the reduction one ends up with 41 master integrals for which an
explicit result is needed. Nine integrals are quite simple and can
essentially be obtained from the one- and two-loop results. 14
integrals contain a massless one-loop diagram which can be integrated
out leading to a two-loop integral with an exponent depending on the
space-time dimension $d$. These integrals are already quite
involved and have been presented in Ref.~\cite{Smirnov:2010gi}. The remaining
18 integrals are genuinely of three-loop order and involve a nontrivial
calculation to obtain their result. All but three $\epsilon=(4-d)/2$
coefficients could be computed analytically; the corresponding
analytical results have been presented in Ref.~\cite{Smirnov:2010zc}. 
The three missing coefficients are known with a 
numerically precision sufficient for all foreseeable applications.


\section{Static potential to three loops}

Let us finally present the result for the static potential. We refrain
from analytical results which can be found in
Refs.~\cite{Smirnov:2008pn,Smirnov:2009fh} but immediately show
$V(|{\vec q}\,|)$ in numerical from:
\begin{eqnarray}
  V(|{\vec q}\,|)&=&-{4\pi C_F\alpha_s(|{\vec q}\,|)\over{\vec q}\,^2}
  \Bigg[1+\frac{\alpha_s}{\pi}\left(2.5833 - 0.2778 n_l\right)
%    \nonumber\\&&\mbox{}
    +\left(\frac{\alpha_s}{\pi}\right)^2\left(28.5468 - 4.1471 n_l 
    + 0.0772 n_l^2 \right)
    \nonumber\\&&\mbox{}
    +\left(\frac{\alpha_s}{\pi}\right)^3\left(
    209.884(1) -51.4048 n_l 
%    \right.\nonumber\\&&\left.\mbox{}
    + 2.9061 n_l^2
    - 0.0214 n_l^3\right)
    +\cdots\Bigg]\,,
  \label{eq::Vnum}
\end{eqnarray}
where $\mu^2={\vec q}\,^2$ has been adopted in order to suppress the 
infrared logarithm and the 
ellipses denote higher order terms in $\alpha_s$.
It is interesting to note that
the term ``209'' in the three-loop coefficient 
receives a large contribution (``211'') from the term with colour
factor $C_A^3$ whereas
the new colour structure $d_F^{abcd}d_A^{abcd}$
only contributes with a coefficient ``$-2$''.
From Eq.~(\ref{eq::Vnum}) we observe at \mbox{one-,} two- and three-loop
order a large screening of the non-fermionic contributions by the $n_l$ terms
which is
most prominent in the three-loop coefficient for $n_l=5$.

In Tab.~\ref{tab::a123} we show the numerical evaluation of the square bracket
of  
Eq.~(\ref{eq::Vnum}) for the charm, bottom and top quark case, i.e. for
$n_l=3,4$ and~5, adopting the appropriate values of $\alpha_s$.
For charm the three-loop corrections are almost as big as the one- and
two-loop contributions whereas for bottom the three-loop contribution
is already a factor of four smaller than the two-loop one.
In the case of the top quark one observes a good convergence: the
three-loop term is already a factor ten smaller than the two-loop
counterpart. 

\begin{table}[t]
  \begin{center}
    \begin{tabular}{c|l|l|l|l}
      $n_l$ & $\alpha_s^{(n_l)}$ & 1 loop & 2 loop & 3 loop \\
      \hline
      3 & 0.40 & 0.2228 & 0.2723 
      & 0.1677 \\
      4 & 0.25 & 0.1172 & 0.08354 
      & 0.02489\\
      5 & 0.15 & 0.05703 & 0.02220 
      & 0.002485
    \end{tabular}
    \caption{\label{tab::a123}Radiative corrections to the potential
      $V(|{\vec q}\,|)$ where the tree-level result is normalized to 1.
      In the second column we also provide the numerical value of
      $\alpha_s$ corresponding to the soft scale where $\mu\approx m_q\alpha_s$
      ($m_q$ being the heavy quark mass).}
  \end{center}
\end{table}

To summarize, the three-loop corrections to the static heavy quark
potential are available and can now be used for the 
prediction of the top quark threshold production at a future linear
collider with third-order accuracy, 
for the precise extraction of the bottom quark mass from
$\Upsilon$ sum rules, and for the comparison of the potential with
results obtained on the lattice in order to gain insight to the validity of
perturbation theory.


\begin{thebibliography}{99}
  \input{ichep10_a3_ref.tex}
\end{thebibliography}

\end{document}

